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Basically I was trying to achieve this: impose an arbitrary image to a pre-defined uneven surface. (See examples below).
I do not have a lot of experience with image processing or 3D algorithms, so here is the best method I can think of so far:
Problems/questions I have:
Your process is essentially correct. If you work pixel by pixel, rather than trying to use triangles, you'll avoid some of the problems you're facing. Scan across the pixels in target bitmap, and apply the local transformation based on the cell you're in to determine the coordinate of the corresponding pixel in the source bitmap. Copy that pixel over.
For a smoother result, you do your coordinate transformations in floating point and interpolate the pixel values from the source image using something like bilinear interpolation.